In Doctor Who, the word "companion" carries a great deal of baggage. There are few things that can't change in the long-running show's universe, but one of the only rules is that the Doctor (usually) doesn't travel alone. The tight focus on the mysterious time lord and whoever happens to be accompanying him typically makes for a pretty tight regular cast, although the revived series has countered this by giving his companions more support through families, boyfriends etc, to varying degrees of success.
While there is usually no mistake as to who gets canon status, this practice has blurred the line between a "real" companion and a minor player somewhat. As a result, there are tons of faces who shared screen time with the lead, and maybe even traveled in the TARDIS, but are still not considered "proper" sidekicks in the manner of Rose, Martha or Donna. My personal definition for a companion has always been someone who begins more than one episode in the TARDIS with the Doctor, but even that leaves out some important figures, like the Brigadier, out of the equation.
In addition, because of the pseudo-anthology nature of the show, any given episode is almost guaranteed to have someone the audience wants to see more of and doesn't get to. Oftentimes, these characters remain in the background because they get killed by the end of their episodes, and that's just story structure for you. But in some cases there was no real reason not to keep them on longer, and a few still have the potential to come back should Steven Moffat ever get tired of feisty, flirtatious women with some sort of chronological mix-up going on. Also, the "official" companion is usually known to the public far in advance and gets a whole episode as an introduction, so it would be nice if the Doctor picked up someone in a little more spontaneous way, for a change.
Here is just a taste of those fleeting souls who had untapped potential. I'm going to stick to just the new (since 2005) series, since a) my Who lists usually don't and b) if I didn't this list could easily be ten times as long. And before you ask, I'm leaving out Madame Vastra, Jenny and Strax from the most recent episodes, as it's been announced that they will return, and it's possible they might join more permanent ranks at some point. Also, mucho spoilers for those who worry about such things.
10) Jabe Ceth Ceth Jafe, "The End of the World"
Two words: tree lady. Plus, although the Doctor has had non-earthlings in his crew before, very rarely have we seen him with non-humans. I'm sure it would have broken the makeup budget and been infeasible for a variety of reasons. Still. Tree lady.
9) Thomas Kincade Brannigan, "Gridlock"
Along similar lines, how cool would it have been to have a cat person? And he's played by Father Dougal, aka Ardal O'Hanlon. True, this would make Doctor Who only the second British sci-fi show to feature a felis sapien in the main cast. But Brannigan had an upbeat spirit, and was a religious and family man, even if a good chunk of his life was spent in a traffic jam in New New York. I still feel like Gridlock is an underrated episode, and Brannigan may not be the main reason why, but he's still pretty fun to watch, and at least he doesn't sing. We'll just overlook the fact that he might have been named after a dubious American painter.
8) Lynda Moss, "Bad Wolf"/"The Parting of the Ways"
The Ninth Doctor's sole season is interesting in that it contained Adam, the fake-out companion who seemed like he'd be joining but got kicked out of the TARDIS after he went and got a hatch installed in his skull (long story). But then Captain Jack came and joined the crew, proving that series head Russell T. Davies wasn't throwing out the idea of a two-companion system for good.
So there may have been a little genuine curiosity when future Big Brother contestant Lynda, played by Jo Joyner, started tagging along during the battle onboard the Game Station, formerly Satellite 5. Would she make it out alive? Well, she's a fairly sweet and innocent person and the Doctor promises he'll protect her, so in retrospect it's not a huge shock that she wasn't long for this universe. But it would have been nice to see more of her and her double ponytail, remniscent of a more restrained Zenon: Girl of the 21st Century. Plus, the hints of jealousy from Rose might have made for a more compelling dynamic.
7) Guido, "The Vampires of Venice"
A decent man who sacrificed everything for the sake of his daughter, Guido was the sort of person who was such a good person you knew he was going to snuff it. This is one episode that, like "Dinosaurs on a Spaceship,"manages to transcend its title and be a little more than a silly joke. Part of the heft comes from how serious actor Lucian Msamati takes his role, even when wearing a shirt from Rory's stag party. As a regular I'm sure he would have brought up all sorts of "what's that, Doctor?" questions, and probably seems like he would have been dead weight in a Dalek situation. But the Doctor's traveled with the technologically primitive before. Heck, at this point he's even brought Nefertiti along, for God's sake. Guido at least has a better chance of knowing what gunpowder is.
6) Dr. Nasreen Chaudhry, "The Hungry Earth"/"Cold Blood"
An intelligent, older woman who isn't sassy or unstuck in time somehow, Nasreen obviously wasn't destined to make it into the opening credits. All the same, she was a compelling, sensible person. Plus, she played a significant role in talking down the Silurians and supposedly setting off a chain of events that could lead to an eventual alliance between the humans and the lizard-people.
Her romance with Tony Mack was genuinely sweet instead of sappy, and her decision to stay hibernating with him seemed to be deliberate bait for a sequel at the time, but I didn't mind because she was pretty cool. Now, it seems like this plotline has been somewhat lost in the shuffle, along with a dozen other things that can't keep up with the increasingly ADD-stricken pace of the most recent seasons. You hear that, Steven Moffat and Chis Chibnail? I want more Meera Syal, damnit!